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Jackson theorems for Erdős weights in $$L_p$$ $$(0<p\leq \infty)$$. (English) Zbl 0915.41013
This paper is a the first part of a work on Jackson estimates for the approximation by polynomials, of functions in weighted $$L_p$$, $$0<p\leq\infty$$, where the weights are Erdős type weights. The second part contains inverse theorems and was written by the first author. For some reason the latter appeared earlier, in [J. Approximation Theory 93, No. 3, 349-398 (1998; Zbl 0907.41010]. In this paper the authors establish the Jackson estimates by introducing new moduli of smoothness, based on the Erdős weights and contain two parts. One which is given by the weighted norm of the differences of the function on a suitably defined finite interval, and the other which measures the best of approximation by polynomials of low degrees, on the infinite part of the interval. The above mentioned second paper shows that these moduli are indeed the proper ones. The proofs are quite technical. The authors first approximate the function by piecewise polynomials of low degree as best as possible (as guaranteed by Whitney), and then replace the piecewise polynomials by polynomials of the much higher degree.

MSC:
 41A30 Approximation by other special function classes 41A10 Approximation by polynomials
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References:
 [1] Brudnyi, Yu.A., Approximation of functions by algebraic polynomials, Izv. akad. nauk SSSR ser. math., 32, 780-787, (1968) · Zbl 0165.38702 [2] Clunie, J.; Kövari, T., On integral functions having prescribed asymptotic growth, II, Canad. J. math., 20, 7-20, (1968) · Zbl 0164.08602 [3] Damelin, S.B., Converse and smoothness theorems for erdős weights in lp(0
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